The enveloping algebra of the Lie superalgebra osp(1, 2)

Pinczon, Georges

doi:10.1016/0021-8693(90)90265-p

Cited by 49 publications

(47 citation statements)

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“…After recalling the definition of second order Casimir operators for Lie superalgebras [20, p. 57] (also see [1,33,32,37,38] and references therein for detailed descriptions of Casimir elements), we specialize to pgl(p + 1|q) ∼ = sl(p + 1|q) (thus assuming q = p + 1). We consider the Casimir operators C and C associated with the representations L and L on S δ .…”

Section: Casimir Operatorsmentioning

confidence: 99%

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Mathonet

Radoux

2011

Lett Math Phys

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Abstract. We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n) ∼ = sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. MSC(2010): 17B66, 58A50

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Section: Casimir Operatorsmentioning

confidence: 99%

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Mathonet

Radoux

2011

Lett Math Phys

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“…Finally, the classification of Prim U(s) and Prim U(h) being well known (e.g., [16]) we obtain a classification of Prim U(g).…”

Section: H((g) Is a Complete Set Of Representations Of Gmentioning

confidence: 78%

“…For simple Lie superalgebras, the situation is more involved: for instance, the ideal U(h)/(L+l/4), of U{h), is minimal primitive, but not generated by its intersection with Z(h) ([16]). Nevertheless, it is the only ideal of this type in U(h) note that it comes from the metaplectic representation [16]. For U(g), complexity is increasing, since: THEOREM 9 (VI.5).…”

Section: H((g) Is a Complete Set Of Representations Of Gmentioning

confidence: 99%

“…The first case is exactly the series osp(l, 2ή), which are also the only semi-simple simple superalgebras [8]. The simplest model of this case is h = osp(l, 2) U(h) was completely studied in [16], including explicit computation of Prim U(h). The simplest model of the second case is g = sl (2,1), and the purpose of the present paper is a complete study of U(g).…”

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confidence: 99%

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The structure of sl(2,1)-supersymmetry: irreducible representations and primitive ideals

Arnal¹,

Amor²,

Pinczon³

1994

Pacific J. Math.

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We give a detailed study of the enveloping algebra of the Lie superalgebra sl(2, 1), including classification of irreducible HarishChandra modules, completeness of finite dimensional irreducible, explicit computation of center, and classification of primitive ideals.Introduction and main results. Lie superalgebras are important both in physics and in mathematics [5]. In physics, they are used e.g. to unify fermions and bosons in a unique picture (one irreducible representation of the structure) via supersymmetry. In mathematics, their enveloping algebras provide a class of very interesting noetherian algebras. Much information is known about enveloping algebras of Lie algebras (e.g., [4]), but for superalgebras there is a lot to do (see e.g.[2] for a pioneering work, and [13] for a very nice survey of results obtained up to now). Let us restrict to the simple case; then a natural distinction does appear between simple superalgebras with an enveloping algebra which is a domain and others. The first case is exactly the series osp(l, 2ή), which are also the only semi-simple simple superalgebras [8]. The simplest model of this case is h = osp(l, 2) U(h) was completely studied in [16], including explicit computation of Prim U(h). The simplest model of the second case is g = sl(2, 1), and the purpose of the present paper is a complete study of U(g). We shall give a classification of irreducible Harish-Chandra modules, a detailed computation of the center Z(g) of U(g), and a classification of Prim U(g).Let us recall known results: finite dimensional irreducible representations of g = sl(2, 1) are known [18], and also unitary irreducible are classified ([6], [7]). Moreover, finite dimensional representations provide a complete set of representations [2], but are generally not fully reducible.A fundamental result of our paper is the fact that finite dimensional irreducible provide a complete set, because of information that can be deduced on U(g). Actually, we deduce an explicit determination of the center Z(g), which shows that Z{g) is not a finitely

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“…When r = 1 the decomposition of the adjoint representation is given in [P,Lemma 7.4.1]. Part 1) of the theorem also follows from arguments in [PS,Section 3].…”

Section: Lemmamentioning

confidence: 94%

Some Lie superalgebras associated to the Weyl algebras

Musson

1999

Proc. Amer. Math. Soc.

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Abstract. Let g be the Lie superalgebra osp(1, 2r). We show that there is a surjective homomorphism from U (g) to the r th Weyl algebra Ar, and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of g on Ar and use this to show that if Ar is made into a Lie superalgebra using its natural Z 2 -grading, then We work throughout over an algebraically closed field k of characteristic zero. If g is a simple Lie algebra different from s (n), Joseph shows in [J2], that there is a unique completely prime ideal, J 0 whose associated variety is the closure of the minimal nilpotent orbit in g * . When g is the symplectic algebra g = sp(2r), this ideal may be constructed as follows. It is well known that the symmetric elements of degree two in the r th Weyl algebra A r form a Lie algebra isomorphic to sp(2r) [D, Lemma 4.6.9]. Hence there is an algebra map φ : U (g) −→ A r whose kernel is clearly completely prime and primitive. Since the image of φ has Gel'fand Kirillov dimension 2r, and this is the dimension of the minimal nilpotent orbit in g * by [CM, Lemma 4.3.5], we have ker φ = J 0 . Now if g is a classical simple Lie superalgebra, and U (g) contains a completely prime primitive ideal different from the augmentation ideal, then g is isomorphic to an orthosymplectic algebra osp(1, 2r) (Lemma 1). We observe that if g = osp(1, 2r), then there is a surjective homomorphism U (g) −→ A r whose kernel J satisfies J ∩ U (g 0 ) = J 0 . The existence of this homomorphism has previously been shown in the Physics literature; see, for example, [F, pages 55 and 170]. It follows that g acts via the adjoint representation on A r , and we determine the decomposition of this representation explicitly.This turns out to be a useful setting in which to study the Lie structure of certain associative algebras. A result of Herstein [He]

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The enveloping algebra of the Lie superalgebra osp(1, 2)

Cited by 49 publications

References 8 publications

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

The structure of sl(2,1)-supersymmetry: irreducible representations and primitive ideals

Some Lie superalgebras associated to the Weyl algebras

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